Key estimates

Remark 4.1.1. Consider a family of metrics ω(t, x) for (t, x) ∈ R. For t ∈ [0, T), t ∈ (T, T⁰], we require ω(t) to be smooth at t in the usual sense, in M, N respectively.

On the other hand, if (t, x) = (T, x) ∈ T ×(N \ {y₀}) ∼= T ×(M \E), then we choose a sufficiently small neighborhood U of x in M \E and we consider ω as a metric on (T −δ, T +δ)×U for someδ > 0 via the mapπ. We say ω(t) is smooth at (T, x) if ω(t) is smooth at (T, x) in (T −δ, T +δ)×U. In the same way, we can define what it means for ω(t) to satisfy a PDE at an arbitrary point of R.

In this sense, we can continue the Chern-Ricci flow starting at a Gauduchon metric until we contract all finitely many (−1)-curves on a given non-K¨ahler compact complex surface and eventually reach a minimal surface. Additionally, that (N, ω(t)) converge to (N, d_T) in the Gromov-Hausdorff sense can be shown by the same way as in section 6 in [59] with using Lemma 3.4 and Lemma 3.5 in [70].

The result of Theorem 3.1.1 indicates that the requirement of the cohomology classes for the convergence of the Chern-Ricci flow:

(†) [ω₀] +T c^BC₁ (K_M) = [π^∗ωˆ_N]

holds under the assumptions in Theorem 3.1.1. Then, we can say that the Chern-Ricci flow performs a canonical surgical contraction in the sense of Definition 1.2.5:

Theorem 4.1.1. Let ω(t) be a smooth solution of the Chern-Ricci flow on M starting at ω₀ for t ∈[0, T), 0 < T <∞. Assume that ω(t) is non-collapsing at T. Suppose that there exists a blow-down map π :M →N contracting the only one (−1)-curve E to the point y₀ ∈ N. Then the Chern-Ricci flow ω(t) performs a canonical surgical contraction with respect to the data E, N and π.

As considering the definition in [70], in order to say that g(t) performs a canonical surgical contraction in the sense of [70], it additionally requires to show that (N, dT) is the metric completion of (N\ {y₀}, d_gT), where these notations are the same as in Defenition 1.2.5. It only suffices to prove that d_gT = d_T|N\{y₀}. In the K¨ahler case (cf. [60]), this can be shown with using the fact that any K¨ahler metrics are locally given by K¨ahler potentials. Hence we expect that it requres new techniques in the non-K¨ahler case.

onM is written with using u⁰₀ and ˆω_N in the following way:

ω(t) = ˆω_t+√

−1∂∂ϕ¯ _t,

where ˆω_t:= _T¹((T −t)ω₀+tπ^∗ωˆ_N) and ϕ_t solves the parabolic Monge-Amp`ere flow:

(M AF) ∂

∂tϕ_t= logω(t)²

Ω , ϕ_t|_t=0 = 0, with Ω =ω²₀e^u

0 0

T . Note thatϕ_tis uniformly bounded from above and below onM×[0, T).

One can show that the two flows (CRF) and (M AF) are essentially equivalent:

If ϕ_t solves (M AF), then taking √

−1∂∂¯of (M AF) shows that

√−1∂∂¯∂

∂tϕ_t

= √

−1∂∂¯log ω(t)² ω²₀ − 1

T

√−1∂∂u¯ ⁰₀

= −Ric(ω(t)) + Ric(ω₀)− 1

Tπ^∗ωˆ_N + 1

T(ω₀ −T Ric(ω₀))

= −Ric(ω(t))− ∂

∂tωˆ_t,

which implies we have (CRF). Conversely, if ω(t) solves (CRF), then we have

∂

∂t(ω(t)−ωˆt) = −Ric(ω(t))− 1

Tω0+ 1 Tπ^∗ωˆN

= √

−1∂∂¯

log ω(t)² ω²₀ − 1

Tu⁰₀

= √

−1∂∂¯logω(t)² Ω

so if we choose ϕ_t to solve (M AF), which is an ODE int for fixed point on M, then we obtain

∂

∂t(ω(t)−ωˆ_t−√

−1∂∂ϕ¯ _t) = 0 so that indeed ω(t) = ˆω_t+√

−1∂∂ϕ¯ _t and ϕ_t satisfies (M AF).

Since the positive current ω(T), which is smooth onM⁰, can be written by ω(T) =π^∗ωˆ_N +√

−1∂∂ϕ¯ _T ≥0,

where ϕ_T is a bounded function satisfiesϕ_T|_E ≡constant since we have

√−1∂∂ϕ¯ _T|_E =ω(T)|_E ≥0

and then we apply the strong maximum principle. Hence, from the properties of the blow-down map π, there exists a bounded function ψ_T on N, smooth on N \ {y₀}, with ϕ_T =π^∗ψ_T. Especially, we have ψ_T ∈PSH(N \ {y₀},ωˆ_N)∩C⁰(N\ {y₀}).

We here define a ∂∂-closed positive (1,¯ 1)-currentω⁰ onN by ω⁰ := ˆω_N +√

−1∂∂ψ¯ _T ≥0,

which is smooth and positive onN \ {y₀} and satisfies π^∗ω⁰ =ω(T). We have 0≤ ω⁰²

ˆ

ω²_N ∈L^p(N,ωˆ_N²)

for some p >1 sufficiently close to 1 (cf. [59, Lemma 5.4]) and ω⁰² >0 on N \ {y₀}. We consider the equation

ω⁰² = ˆfωˆ²_N onN \ {y0}, where we put ˆf := ^ω_ωˆ2⁰²

N. We normalize ψT such that sup_N\{y0}ψT = 0. Then ψ_T is a unique continuous ˆω_N-psh solution of the equation ω⁰² = ˆfωˆ_N² on N\ {y₀}.

We would like to construct a solution of the Chern-Ricci flow on N starting at the metric ω⁰. We fix a smooth d-closed (1,1) form −χ∈c^BC₁ (N). Then there exists T⁰ > T sufficiently close toT such that for all t∈[T, T⁰], the following (1,1)-form

ˆ

ω_t,N := ˆω_N + (t−T)χ

is Gauduchon. We also fix a smooth volume form Ω_N on N satisfying Ric(Ω_N) = −√

−1∂∂¯log Ω_N =−χ∈c^BC₁ (N).

For ε >0 sufficiently small, and A sufficiently large, define a family of volume forms Ω_ε on N by

Ω_ε:= (π|⁻¹_M0)^∗|s|^2A_h ω(T −ε)²

|s|^2A_h +ε

+εΩ_N

onN\ {y₀}, and Ω_ε|_y0 =εΩ_N|_y0 wheresis a holomorphic section with E = (s), where (s) is a principal divisor defined bys, andhis a smooth Hermitian metric on the holomorphic line bundle [E] associated to the effective divisorE respectively. Note that Ω_ε is smooth on N \ {y₀}. By choosing A sufficiently large , the volume form Ω_ε lies in C^l(N) for a fixed large constant l. And note that Ω_ε converges to ω⁰² in C^∞ on any compact subsets of N \ {y₀} as ε → 0. Now, for each ε > 0, by the theorem of Tosatti and Weinkove (Theorem 2.5.5), there exist a unique constantC_ε∈R>0 and a unique function ψ_{T ,ε} ∈ C^k(N)∩C^∞(N \ {y₀}) for some positive integer k with sup_N\{y

0}ψ_{T ,ε} = 0 such that

(ˆω_N +√

−1∂∂ψ¯ _{T ,ε})² =C_εfˆ_εωˆ_N², where we put ˆf_ε := _ω^Ω_ˆ2^ε

N

. Since we have 0 ≤ ^ω_ωˆ2⁰² N

∈ L^p(N,ωˆ²_N) for some p > 1, ˆf_ε’s are uniformly bounded in L^p(N,ωˆ_N² ) for some p > 1. Notice that we can freely rase k by increasing l and A. The constants C_ε > 0 satisfy that C_ε → 1 as ε → 0. We define the admissible function h(x) := CC_ε⁻¹||fˆ_ε||⁻¹_Lp(N,ˆω²_N)exp(ax) for some uniform constants C, a > 0. Then (ˆω_N +√

−1∂∂ψ¯ _{T ,ε})² satisfies (♣)_ωˆN from Proposition 2.5.3, and then we may apply Proposition 2.5.4.

Lemma 4.2.1. There exists a uniform positive constant C =C(||f_ε||_L^p, N,ωˆ_N)>0 such

that 1

C ≤C_ε ≤C.

Proof. Fix 0< δ <1. Define S_ε:= inf_Nψ_{T ,ε} and δ₀ := 1

3min{δ², δ³

16B,4(1−δ)δ²,4(1−δ) δ³ 16B}.

Then for 0< s, t < δ0, we have (Remark 2.5.1) t²cap_ωˆ

N({ψ_{T ,ε}< S_ε+s}) ≤ C Z

{ψ_{T ,ε}<Sε+s+t}

C_εfˆ_εωˆ_N²

≤ CC_ε||fˆ_ε||_Lp(N,ˆω_N²)Vol_ωˆN({ψ_{T ,ε}< S_ε+s+t})^1q, where ¹_p +¹_q = 1. Hence for fixed 0 < s=t < δ0, we obtain

cap_ωˆ

N({ψ_{T ,ε}< S_ε+s}) ≤ CC_ε

sⁿ ||fˆ_ε||_Lp(N,ˆω²_N))Vol_ωˆN({ψ_{T ,ε} < S_ε+ 2s})^1q

≤ C⁰C_ε

sⁿ Vol_ωˆN(N)^1q =:C_εC₁s⁻ⁿ

for some uniform constantC⁰ >0, where we used that||fˆ_ε||_Lp(N,ˆω²_N)) is uniformly bounded from above. and then from Proposition 2.5.4,

s≤κ(cap_ωˆN({ψ_T,ε < S_ε+s}))≤κ(C_εC₁s⁻ⁿ).

Since limx→0⁺κ(x) = 0, C_ε must be uniformly bounded away from 0.

Since ˆf_ε → fˆin L¹(N,ωˆ_N² ), we also have ˆf

1

ε2 → fˆ¹² in L¹(N,ωˆ²_N). Since we have R

N fˆ¹²ωˆ_N² >0, for ε sufficiently small, we obtain Z

N

fˆ

1

ε2ωˆ_N² > 1 2

Z

N

fˆ¹²ωˆ_N² >0.

By the pointwise arithmetic-geometric means inequality implies that (ˆωN +√

−1∂∂ψ¯ T ,ε)∧ωˆN ≥(ˆω_N +√

−1∂∂ψ¯ _{T ,ε})² ˆ

ω_N²

¹₂ ˆ

ω²_N = (Cεfˆε)¹²ωˆ_N². It follows that for sufficiently smallε,

C

1

ε2 ≤ 2

R

Nfˆ¹²ωˆ_N² Z

N

(ˆω_N +√

−1∂∂ψ¯ _{T ,ε})∧ωˆ_N = 2 R

Nfˆ¹²ωˆ²_N Z

N

ˆ ω²_N, where we used the Stokes theorem and that ˆωN is Gauduchon.

Suppose that there exists a subsequence C_εk → c 6= 1 as k → ∞. Consider the equation

(ˆωN +√

−1∂∂ψ¯ T,ε_k)² =Cε_kfˆε_kωˆ_N².

Then since the family{ψ_{T ,εk} ∈PSH(ˆω_N)∩C⁰(N\ {y₀}); sup_N\{y0}ψ_{T ,εk} = 0}is relatively compact inL¹(N\{y₀},ωˆ_N²), after passing a subsequence, still writeψ_{T ,εk}, sinceC_εkfˆ_εk are uniformly bounded inL^p(N,ωˆ²_N) for somep >1 sufficiently close to 1, we have that{ψ_T,εk}

is a Cauchy sequence in C⁰(N \ {y₀}) (Corollary 2.5.3). This means thatψ_T,εk →ψ⁰_T for some ψ_T⁰ ∈PSH(ˆω_N)∩C⁰(N\ {y₀}) inC⁰(N\ {y₀})-topology with sup_N\{y

0}ψ⁰_T = 0. By the Bedford-Taylor convergence theorem (Theorem 2.5.1), we obtain by taking the limit onN \ {y₀}, since ψ_T is the unique solution of the equation (ω⁰)² = ˆfωˆ²_N,

(ω⁰)² = (ˆω_N +√

−1∂∂ψ¯ _T⁰ )² =cfˆωˆ²_N =c(ω⁰)², which is a contradiction. Hence we conclude thatC_ε →1 as ε→0.

For the following two equations (ˆω_N +√

−1∂∂ψ¯ _T)² = ω⁰² ˆ

ω_N² ωˆ_N², (ˆω_N +√

−1∂∂ψ¯ _{T ,ε})² = C_εΩ_ε ˆ ω²_N ωˆ_N², we apply the stability theorem:

Proposition 4.2.1. ([50, Theorem A.])Let (Xⁿ, ω) be a compact n-dimensional Hermi-tian manifold. Let 0 ≤ f, g ∈ L^p(X, ωⁿ), p > 1, be such that R

Xf ωⁿ > 0, R

Xgωⁿ > 0.

Consider two continuous ω-psh solutions of the complex Monge-Amp`ere equation (ω+√

−1∂∂u)¯ ⁿ =f ωⁿ, (ω+√

−1∂∂v)¯ ⁿ=gωⁿ with sup_X u= sup_Xv = 0. Assume that f that

f ≥c₀ >0

for some uniform positive constantc₀ >0. Fix 0< α < _n+1¹ . Then, there exists a positive constant C =C(c₀, α,||f||_L^p,||g||_L^p)>0 such that

||u−v||L^∞ ≤C||f−g||^α_Lp.

Now we apply Proposition 4.2.1 for X = N \ {y₀}, u = ψ_{T ,ε}, v = ψ_T, f = ^C_ωˆ^εΩ2^ε N

and g = _ω^ω_ˆ⁰²2

N, since we have 0≤ ^ω_ωˆ2⁰² N,^C_ωˆ^ε2^Ωε

N ∈L^p(N,ωˆ²_N) for somep > 1 and (ω⁰)² >0 onN\{y0}, which indicates that we can choose a uniform constant c₀ >0 independent ofε such that

C_εΩ_ε ˆ

ω²_N ≥c0 >0.

Then we obtain, for arbitrary fixed 0< α < ¹₃,

||ψ_T,ε−ψ_T||_L^∞_(N\{y₀}) ≤C

C_εΩ_ε ˆ

ω_N² − ω⁰² ˆ ω_N²

α

L^p(N\{y0}).

Hence ψ_{T ,ε} converges to ψ_T on N \ {y₀} in L^∞-topology as ε → 0. It follows that we obtain that as ε→0,

||ψ_{T ,ε}−ψ_T||_L^∞_(N) = sup

N

|ψ_{T ,ε}−ψ_T|= sup

N\{y0}

|ψ_{T ,ε}−ψ_T| →0.

Thus we have that ψ_T ∈PSH(N,ωˆ_N)∩C⁰(N) with sup_Nψ_T = 0.

With using the regularity of the functions ψ_{T ,ε}, we can show the following result for solutionsϕ_ε=ϕ_ε(t) of the parabolic complex Monge-Amp`ere equations

∂

∂tϕ_ε = log(ˆω_t,N +√

−1∂∂ϕ¯ _ε)²

Ω_N , for t ∈[T, T⁰], ϕ_ε|_t=T =ψ_{T ,ε}, which is equivalent to the Chern-Ricci flow

∂

∂tωε(t) =−Ric(ωε(t)), for t∈[T, T⁰], ωε(T) = ωT ,ε

where ω_ε=ω_ε(t) := ˆω_t,N +√

−1∂∂ϕ¯ _ε and ω_{T ,ε} := ˆω_N +√

−1∂∂ψ¯ _T,ε. Then we obtain the following results as in [59].

Proposition 4.2.2. ([59, Proposition 5.1])

There exists a function ϕ∈C⁰([T, T⁰]×N)∩C^∞((T, T⁰]×N) such that

(1) ||ϕ_ε||_L^∞ ≤C for some uniform constantC > 0 for allε >0 sufficiently small.

(2) ϕ_ε →ϕin L^∞([T, T⁰]×N).

(3) The convergenceϕ_ε→ϕis C^∞ on compact subsets of (T, T⁰]×N. (4) ϕis the unique solution of

∂

∂tϕ= log(ˆω_t,N +√

−1∂∂ϕ)¯ ²

Ω_N , ϕ|_t=T =ψ_T for t∈(T, T⁰] in the space C⁰([T, T⁰]×N)∩C^∞((T, T⁰]×N).

We take advantage of the following result:

Proposition 4.2.3. ([54,Theorem 1.1, Corollary 1.2.])

Fixr with 0< r <1. Letω(t) solve the Chern-Ricci flow fort∈[0, T₀],T₀ <∞, starting atω₀; a Hermitian metric on a Hermitian manifold M, in a neighborhood ofB_r, which is the ball of radius r at the origen in Cⁿ, fort ∈[0, T₀]. Assume R >1 satisfies

1

Rω₀ ≤ω(t)≤Rω₀ onB_r×[0, T₀].

Then there exist positive constants C, α, β depending only on ω₀ such that (1) |∇⁰ω|²_ω ≤ ^CR_r2^α on B^r

2 ×[0, T0],where ∇⁰ is the Chern connection ofω0. (2) |Rm|²_ω ≤ ^CR_r4^β on B^r

4 ×[0, T₀], for Rm the Chern curvature tensor of ω.

(3) For any δ > 0 with 0 < δ < T₀, there exist constants C_m, α_m and γ_m for m = 1,2,3, . . . depending only on ω₀ and δ such that

|(∇⁰

R)^mω|²_ω

0 ≤ C_mR^αm

r^γm onB^r

8 ×[δ, T₀], where ∇⁰_R is the Levi-Civita covariant derivative associated to ω₀.

For the metric ω_{T ,ε}, which is smooth away from y₀, we have the following estimate:

Lemma 4.2.2. For all sufficiently smallε >0, there exist positive uniform constants C, α, independent of such ε, such that on N \ {y₀},

|s|^2α_h

C ωˆ_N ≤ω_{T ,ε}≤ C

|s|^2α_h ωˆ_N.

Fix a large positive integer K. Then for each integer 0 ≤ k ≤ K there exist C_k, α_k > 0 such that

|( ˆ∇^N_R)^kω_{T ,ε}|²_ωˆ

N ≤ C_k

|s|^2α_h ^k, where ˆ∇^N

R denotes the real Levi Civita covariant derivative with respect to the metric ˆω_N. Remark 4.2.1. We identify a small neighborhood ofy₀ ∈Y with a small ballB centered at the origin ofC². From the property of the blow-down mapπ, we identify viaπthe sets π⁻¹(B \ {0}) and B \ {0}, and for the various functions and (1,1)-forms on these sets.

For instance, we write|s|^2α_h as (π|⁻¹_M\E)^∗(|s|^2α_h ) for simplicity.

Proof. We fix arbitrary sufficiently small real numbersε₀ andδ withε₀ > δ >0 and considerε ∈[δ, ε₀]. From the definition of Ω_ε, ω_{T ,ε}² =C_εΩ_ε, together with

(∗) |s|^2η_h

C ω₀ ≤ω(t)≤ C

|s|^2η_h ω₀

for t∈[0, T) and for some uniform positive constants C, η, where ω(t) = ˆω_t+√

−1∂∂ϕ¯ _t, ˆ

ωt= _T¹((T −t)ω0+tπ^∗ωˆN) (cf. [54, Theorem 1.1] and [59 ,Lemma 2.5]), we have, for F_ε:= logω_{T ,ε}²

ˆ

ω²_N = logC_εΩ_ε ˆ ω_N² ,

∆Fˆ _ε =

∆ logˆ ω_T,ε² ˆ ω_N²

=

−tr_ωˆNRic(Ω_ε) + tr_ωˆNRic(ˆω_N) ≤ C

|s|^2β_h

for some uniform constants β, C > 0, where ˆ∆ for the Laplacian with respect to ˆg (cf.

[59,Lemma 5.3]).

By choosing local coordinates (z₁, z₂), then locally we will writeω_{T ,ε}=√

−1g_i¯jdzⁱ∧d¯z^j, ˆ

ωN = √

−1ˆg_i¯jdzⁱ ∧ d¯z^j. We write ∇, ˆ∇ for the Chern connections associated to g, ˆg respectively. We also write ∆ for the Laplacian of g.

Then we can estimate (cf. [72, Proposition 3.1])

∆ log tr_gˆg ≥ − 2

(tr_ˆgg)²Re

g^k¯lTˆ_kiⁱ∇ˆ¯ltr_gˆg

−Ctr_ggˆ− 1

tr_ˆggtr_gˆRic(g)

≥ − 2

(tr_ˆgg)²Re

g^k¯lTˆ_kiⁱ∇ˆ¯ltr_gˆg

−Ctr_ggˆ− 1 tr_ˆgg

C

|s|^2β_h

for some uniform constantsβ, C > 0, where ˆT is the torsion tensor of ˆg and Ric(g) is the second Ricci tensor with respect to g and we used that |∆Fˆ _ε| ≤ ^C

|s|^2β_h for estimating

tr_gˆRic(g) ≤ C

|s|^2β_h

for some constantC > 0. Here note that we will write Ric₁, Ric for the first Ricci tensor and the second Ricci tensor with respect to g:

(Ric₁)_k¯l =g^i¯jR_i¯jk¯l, Ric_i¯j =g^k¯lR_i¯jk¯l. We define

Q:= log tr_ˆgg−Aψ˜_T,ε+ 1 ψ˜T ,ε+ ˜C

for sufficiently large A, α > 0, where ˜ψ_{T ,ε} := ψ_T,ε− _A¹ log|s|^2α_h and ˜C is a constant such that ˜ψ_T,ε + ˜C ≥ 1. Since Q → −∞ as x → y₀, we may assume that Q achieves its maximum at a pointx₀ ∈N\ {y₀}. Note that we may assume that tr_gˆg ≥1 and |s|^2β_h ≤1 atx₀.

At the point x₀, we have 1

tr_ˆgg∇ˆ¯ltr_gˆg =

A+ 1

( ˜ψ_{T ,ε}+ ˜C)²

∂¯lψ˜_{T ,ε}. We compute at x₀,

0≥∆Q ≥ − 2

(tr_ˆgg)²Re

g^k¯lTˆ_kiⁱ∇ˆ¯ltr_ˆgg

−Ctr_ggˆ− C

|s|^2β_h

−

A+ 1

( ˜ψT,ε+ ˜C)²

tr_g(g −gˆ+ α

AR_h) + 2|∂ψ˜_T,ε|²_g ( ˜ψT ,ε+ ˜C)³

≥ (−C+Ac₀)tr_gˆg− C_A

|s|^2β_h

for some constantCA>0, where we used that for an arbitrary fixed constant 1> c0 >0, ˆ

g−_A^αR_h ≥c₀ˆgfor sufficiently largeA,R_h is the curvature of the smooth Hermitian metric h given locally by

Rh =−√

−1∂∂¯logh.

Remark that we have√

−1∂∂¯logh=√

−1∂∂¯log|s|²_h away fromy₀. And we also estimated in the following way:

(f)

2

(trˆgg)²Re

g^k¯lTˆ_kiⁱ ∇ˆ¯ltr_gˆg ≤

2 trgˆgRe

A+ 1

( ˜ψ_{T ,ε}+ ˜C)²

g^k¯lTˆ_kiⁱ ∂¯lψ˜_{T ,ε}

≤ |∂ψ˜T ,ε|²_g

( ˜ψ_{T ,ε}+ ˜C)³ +CA²( ˜ψ_T,ε+ ˜C)³ tr_gˆg (trˆgg)².

Since we may assume that (tr_ˆgg)² ≥A²( ˜ψ_{T ,ε}+ ˜C)³, we have

− 2 (tr_ˆgg)²Re

g^k¯lTˆ_kiⁱ∇ˆ¯ltrgˆg

≥ − |∂ψ˜_{T ,ε}|²_g

( ˜ψ_{T ,ε}+ ˜C)³ −Ctrgˆg.

If necessary, we again choose a much larger constant A and then we have Ac₀ > C in the estimate above. Therefore, we obtain

tr_gg(xˆ ₀)≤ C

|s|^2β_h . Hence we have

tr_ˆgg(x₀)≤tr_gˆg(x₀)e^Fε ≤ C

|s|^2β_h . Since ψ_{T ,ε} is uniformly bounded, we obtain

Q≤Q(x₀)≤log(C|s|^2(α−β)_h ) +C ≤C

for α sufficiently large so thatα > β and we obtain the desired estimate.

For the higher order estimates for ω_{T ,ε}, we firstly consider the quantity ST ,ε :=|(∇HT ,ε)H_{T ,ε}⁻¹|²_g

where (H_T,ε)ⁱ_l := ˆg^i¯jg_l¯j, ∇is the covariant derivative with respect to ω_{T ,ε}=g and we here write ∆ for the rough Laplacian ofω_T,ε, ∆ =∇^¯k∇_k, where ∇^¯k=g^¯kl∇_l (cf. [52], [54]).

Note that we compute

H_jlⁱ := ((∇_jH_{T ,ε})H_{T ,ε}⁻¹)ⁱ_l = Γⁱ_jl−Γˆⁱ_jl

where Γⁱ_jl, ˆΓⁱ_jl denote the Christoffel symbols of ω_T,ε = g, ˆω_N = ˆg respectively and then we have

S_{T ,ε} =|H|²_g. By commuting ∇ and ¯∇, we obtain

∆H¯ _jlⁱ −∆H_jlⁱ = (Ric₁)_j^rH_rlⁱ + (Ric₁)_l^rH_jrⁱ −(Ric₁)_rⁱH_jl^r

for some constant C >0, where (Ric₁)_j^r is the first Ricci tensor with respect to ω_{T ,ε}. With using the inequality

(∗∗) |s|^2α_h

C ωˆ_N ≤ω_{T ,ε} ≤ C

|s|^2α_h ωˆ_N and the following;

∆H_jlⁱ =∇^¯kRˆ_j¯klⁱ− ∇^k¯R_j¯klⁱ,

where R_j¯klⁱ, ˆR_j¯klⁱ are the Chern curvature tensors of ω_{T ,ε}, ˆω_N respectively.

Here we notice that the Bianchi identities will not hold necessarily for general Hermi-tian manifolds: Let (M, g) be an-dimensional compact Hermitian manifold and let ∇be the Chern connection of g with Christoffel symbols Γ^k_ij and torsion T given by:

Γ^k_ij =g^k¯l∂_ig_j¯l, T_ij^k = Γ^k_ij −Γ^k_ji and

T_ik¯l=T_ik^jg_j¯l= ˆT_ik^jgˆ_j¯l = ˆT_ik¯l

since g_i¯j = ˆg_i¯j+∂_i∂¯jψ_{T ,ε}. There are extra torsion terms in the following identities:

R_i¯jk¯l−R_k¯ji¯l =−∇¯jT_ik¯l

R_i¯jk¯l−R_k¯li¯j =−∇iT¯j¯lk

R_i¯jk¯l−R_k¯li¯j =−∇¯jT_ik¯l− ∇_kT¯j¯li

∇_pR_i¯jk¯l− ∇_iR_p¯jk¯l=−T_pi^rR_r¯jk¯l

∇_q¯R_i¯jk¯l− ∇¯jR_i¯qk¯l =−T_q¯^¯s¯jR_i¯sk¯l. With using the identities above, we then compute

∆ST,ε = g^p¯q∇p∇q¯

g^i¯ag^j¯bgk¯cH_ij^kH_ab^c

= |∇H|¯ ²_g+|∇H|²_g +g^i¯ag^j¯bg_k¯c

∆H_ij^k ·H_ab^c +H_ij^k ·∆H¯ _ab^c

= |∇H|¯ ²_g+|∇H|²_g + 2Re

(−∇_pR_i¯pj¯lg^k¯l+∇_pRˆ_i¯pj^k)H_k^ij +g^i¯ag^j¯bg_k¯cH_ij^k

(Ric₁)^¯r_¯aH_rb^c + (Ric₁)^r¯_¯bH_ar^c −(Ric₁)^c¯_¯rH_ab^r

= |∇H|¯ ²_g+|∇H|²_g +2Re

−( ˆ∇iRicˆ _j¯k−H_ij^rRicˆ _r¯k−∇ˆi∂j∂k¯Fε+H_ij^r∂r∂k¯Fε)H_k^ij +∇i∇p¯Tˆ_pj¯lg^k¯lH_k^ij +∇i∇jTˆ_p¯¯lpg^k¯lH_k^ij + ˆTpi¯sg^r¯sR_r¯pj¯lg^k¯lH_k^ij +( ˆ∇_pRˆ_i¯pj^k−H_pi^rRˆ_r¯pj^k−H_pj^r Rˆ_ipr¯^k+H_pr^kRˆ_i¯pj^r)H_k^ij +g^i¯ag^j¯bg_k¯cH_ij^k

Ric_r¯aH_rb^c −∇ˆ_p¯Tˆ_ps¯ag^s¯rH_rb^c +H_pa^mH_rb^cTˆ_psm¯g^s¯r−∇ˆ_sTˆ_p¯¯apg^s¯rH_rb^c +H_sp^mH_rb^cTˆ_p¯¯amg^s¯r + Ric_r¯bH_ar^c −∇ˆ_p¯Tˆ_ps¯bg^s¯rH_ar^c +H_pb^mH_ar^c Tˆ_psm¯g^s¯r−∇ˆ_sTˆ_p¯¯bpg^s¯rH_ar^c +H_sp^mH_ar^c Tˆ_p¯¯bmg^s¯r

−(Ric_c¯rH_ab^r −∇ˆ_p¯Tˆ_ps¯rg^s¯cH_ab^r +H_pr^mH_ab^rTˆ_psm¯g^s¯c−∇ˆ_sTˆ_p¯¯rpg^s¯cH_ab^r +H_sp^mH_ab^r Tˆ_p¯¯rmg^s¯c)

≥ 1

2|∇H|¯ ²_g+ 1

2|∇H|²_g− C

|s|^2β_h (S

3 2

T,ε+S_T,ε+S

1 2

T ,ε+ 1) since we have Ricr¯a = ˆRicr¯a−∂r∂¯aFε and|∆Fˆ ε| ≤ ^C

|s|^2β_h , where Ric and ˆRic are the second

Ricci curvatures with respect to g and ˆg respectively and we used that T_ijk¯ = ˆT_ij¯k,

∇_pR_i¯pj¯l = ∇_iR_p¯pj¯l−T_pi^rR_r¯pj¯l

= ∇_iR_j¯lp¯p− ∇_i∇_p¯T_pj¯l− ∇_i∇_jT_p¯¯lp−Tˆ_pi¯sg^r¯sR_r¯pj¯l

= ∇_iRicˆ _j¯l− ∇_i∂_j∂¯lF_ε− ∇_i∇_p¯Tˆ_pj¯l− ∇_i∇_jTˆ_p¯¯lp −Tˆ_pi¯sg^r¯sR_r¯pj¯l

= ∇ˆ_iRicˆ _j¯l−H_ij^rRicˆ _r¯l−∇ˆ_i∂_j∂¯lF_ε+H_ij^r∂_r∂¯lF_ε

−∇_i∇_p¯Tˆ_pj¯l− ∇_i∇_jTˆ_p¯¯lp−Tˆ_pi¯sg^r¯sR_rpj¯¯l,

∇_i∇_p¯Tˆ_pj¯l = ∇_i( ˆ∇_p¯Tˆ_pj¯l−H_pl^sTˆ_pj¯s)

= ∇ˆ_i∇ˆ_p¯Tˆ_pj¯l−H_ip^r∇ˆ_p¯Tˆ_rj¯l−H_ij^r∇ˆ_p¯Tˆ_pr¯l− ∇¯iH_pl^sTˆ_pj¯s

−H_pl^s∇ˆ_iTˆ_pj¯s+H_pl^sH_ip^rTˆ_rj¯s+H_pl^sH_ip^rTˆ_pr¯s,

∇i∇jTˆ_p¯¯lp = ∇i( ˆ∇jTˆ_p¯¯lp−H_jp^rTˆ_p¯¯lr)

= ∇ˆ_i∇ˆ_jTˆ_p¯¯lp−H_ij^r∇ˆ_rTˆ_p¯¯lp−H_ip^r∇ˆ_jTˆ_p¯¯lr− ∇_iH_jp^rTˆ_p¯¯lr

−H_jp^r ∇ˆ_iTˆ_p¯¯lr+H_jp^r H_ir^sTˆ_p¯¯ls,

∇¯iH_pl^sTˆ_pj¯sg^k¯lH_k^ij

≤CS_{T ,ε}+ 1

4|∇H|¯ ²_g,

∇_iH_jp^r Tˆ_p¯¯lrg^k¯lH_k^ij

≤CS_T,ε+1 2|∇H|²_g and

2Re( ˆTpi¯sg^r¯sR_rpj¯¯lg^k¯lH_k^ij)

≤CST ,ε+1 4|∇H|¯ ²_g for some constant C >0.

We also compute

∆tr_gˆg = −ˆg^k¯lRic_k¯l

−g^i¯jgˆ^k¯l

Γˆ^p_lj∇ˆ_kg_i¯p+ ˆR_k¯liq¯ˆg^p¯qg_p¯j−Γˆ^p_ki∇ˆ¯lg_p¯j−Γˆ^p_kiΓˆ^q_ljg_p¯q +g^i¯jˆg^k¯lg^p¯q

∇ˆ_kg_i¯q+ ˆΓ^s_kig_s¯q

∇ˆ¯lg_p¯j + ˆΓ^m_ljg_pm¯ +g^i¯jˆg^k¯l

( ˆ∇iTˆ_jl^p)ˆgk¯p+ ( ˆ∇¯lTˆ_ik^p)ˆg_p¯j

−g^i¯jgˆ^k¯l

∇ˆ_iTˆ_jl^p −Rˆ_i¯ls¯jgˆ^s¯p

g_k¯p −g^i¯jgˆ^k¯l

∇ˆ¯lTˆ_ik^p −Rˆ_i¯lk¯qgˆ^p¯q g_p¯j

−g^i¯jgˆ^k¯l

Tˆ_jl^p∇ˆ_ig_k¯p+ ˆT_ik^p∇ˆ¯lg_p¯j

≥ C₁

|s|^2β_h S_{T ,ε}− C

|s|^2β_h (S

1 2

T ,ε+ 1)

for some sufficiently large β >0 and for some constant C₁ >0. Note that we have

|∇tr_ˆgg|²_g ≤ C

|s|^2β_h S_{T ,ε}

for some sufficiently large β >0, and

|∇S_T,ε|²_g ≤2S_T,ε(|∇H|¯ ²_g +|∇H|²_g).

LetBrbe a small ball centered at the origin inC²with radiusr >0. Letρbe a smooth cut off function with suppρ ⊂ B_r and ρ ≡ 1 on B^r

2 such that |∇ρ|²_g +|∆ρ| ≤ _r^C2. We define K := ^C

|s|^2β_h for sufficiently large β >0 and for the constant C >0 in the inequality (∗∗) such that

K

2 ≤K−trˆgg ≤K.

Additionally, for sufficiently large β >0, we may assume that|s|^2β_h << 1 onB_r and then we have

|∇K|_g ≤ C

|s|^3β_h , |∆K|_g ≤ C

|s|^4β_h . For α₀, α₁ >0 sufficiently large withα₀ = 3β < α₁, we define

f :=ρ²|s|^2α_h ¹ S_{T ,ε}

K−tr_ˆgg +A|s|^2α_h ⁰tr_ˆgg.

Note that we may suppose that we have for any sufficiently large α₀ >0,

|∇|s|^2α_h ⁰|_g ≤C|s|^α_h⁰, |∆|s|^2α_h ⁰| ≤C|s|^α_h⁰.

We may assume that f achieves its maximum at a pointx₀ ∈B_r\ {0}. Then, at x₀, we compute

0 = ¯∇f = ∇(ρ¯ ²)|s|^2α_h ¹ S_{T ,ε}

K−tr_ˆgg +ρ²∇(|s|¯ ^2α_h ¹) S_{T ,ε}

K −tr_gˆg +ρ²|s|^2α_h ¹ ∇S¯ _{T ,ε} K−tr_gˆg +ρ²|s|^2α_h ¹ ST ,ε

(K−tr_gˆg)²( ¯∇tr_ˆgg−∇K) +¯ A∇(|s|¯ ^2α_h ⁰)tr_ˆgg+A|s|^2α_h ⁰∇tr¯ _gˆg.

And then, with using this computation, we have at x₀, 0≥∆f = ∆(ρ²)|s|^2α_h ¹ S_{T ,ε}

K−tr_gˆg +ρ²∆(|s|^2α_h ¹) S_T,ε

K−tr_ˆgg +ρ²|s|^2α_h ¹ ∆S_T,ε K−tr_ˆgg +ρ²|s|^2α_h ¹ S_{T ,ε}

(K−tr_ˆgg)²

∆trˆgg−∆K

+A∆(|s|^2α_h ⁰)trˆgg+A|s|^2α_h ⁰∆trˆgg +4Re

ρ∇(ρ)·∇(|s|¯ ^2α_h ¹) S_{T ,ε} K−tr_ˆgg

+ 4Re

ρ∇(ρ)·∇S¯ T ,ε

|s|^2α_h ¹ K −tr_gˆg

+2ρ²Re

∇(|s|^2α_h ¹)·∇S¯ T ,ε

1 K−tr_ˆgg

+ 2ARe

∇(|s|^2α_h ⁰)·∇tr¯ ˆgg

−2Re Atr_ˆgg

K−tr_gˆg(∇tr_ˆgg− ∇K)·∇(|s|¯ ^2α_h ⁰)

−2 A|s|^2α_h ⁰

K−tr_gˆg|∇tr_ˆgg|²_g +2 A|s|^2α_h ⁰

K−tr_ˆggRe

∇tr_ˆgg·∇K¯

We estimate the each term above in the following ways:

|∆(ρ²)||s|^2α_h ¹ S_{T ,ε}

K−trˆgg ≤ C

r²|s|^2α_h ¹S_T,ε K , ρ²|∆(|s|^2α_h ¹)| S_{T ,ε}

K −tr_gˆg ≤Cρ²|s|^2α_h ¹S_{T ,ε} K ,

ρ²|s|^2α_h ¹ ∆S_{T ,ε}

K−tr_ˆgg ≥ ρ²|s|^2α_h ¹ K−tr_ˆgg

1

2|∇H|¯ ²_g+1

2|∇H|²_g− C

|s|^2β_h (S

3 2

T ,ε+ST ,ε+S

1 2

T ,ε+ 1)

≥ ρ²|s|^2α_h ¹ K

|∇H|¯ ²_g+|∇H|²_g

−Cρ²

K |s|^2(α_h ^1−β)(S

3 2

T ,ε+S_{T ,ε}+S

1 2

T ,ε+ 1),

ρ²|s|^2α_h ¹ S_{T ,ε}

(K −tr_gˆg)² +A|s|^2α_h ⁰

∆tr_ˆgg

≥

ρ²|s|^2α_h ¹ S_{T ,ε}

(K −tr_gˆg)² +A|s|^2α_h ⁰ C₁

|s|^2β_h S_{T ,ε}− C

|s|^2β_h (S

1 2

T,ε+ 1)

≥ C⁰4ρ²|s|^2(α_h ^1−β)

K² S_{T ,ε}² +AC₁|s|^2(α_h ^0−β)S_{T ,ε}

−Cρ²|s|^2(α_h ^1−β) K² (S

3 2

T ,ε+S_{T ,ε})−CA|s|^2(α_h ^0−β)(S

1 2

T,ε+ 1), A|∆(|s|^2α_h ⁰)|trˆgg ≤CA|s|^α_H^0−2β,

4 Re

ρ∇(ρ)·∇(|s|¯ ^2α_h ¹) S_{T ,ε} K−trˆgg

≤ C

rK|s|^α_h¹S_{T ,ε}, 4

Re

ρ∇(ρ)·∇S¯ _{T ,ε} |s|^2α_h ¹ K−tr_gˆg

≤ Cρ

rK|s|^2α_h ¹|∇S¯ _{T ,ε}|_g

≤ Cρ

rK|s|^2α_h ¹S

1 2

T ,ε

|∇H|¯ ²_g+|∇H|²_g¹₂

≤ ρ²

2K|s|^2α_h ¹

|∇H|¯ ²_g+|∇H|²_g

+CS_{T ,ε} r²K |s|^2α_h ¹, 2ρ²

Re

∇(|s|^2α_h ¹)·∇S¯ _T,ε 1 K−tr_ˆgg

≤ Cρ²

K |∇(|s|^2α_h ¹)| · |∇S¯ _T,ε|_g

≤ Cρ² K |s|^α_h¹S

1 2

T,ε

|∇H|¯ ²_g+|∇H|²_g¹₂

≤ ρ²

2K|s|^2α_h ¹

|∇H|¯ ²_g +|∇H|²_g + C

KS_{T ,ε}, 2A

Re

∇(|s|^2α_h ⁰)·∇tr¯ _ˆgg

≤CA|s|^α_h⁰|∇tr¯ _ˆgg|_g ≤CA|s|^α_h^0−βS

1 2

T ,ε,

2 Re

Atr_ˆgg

K −tr_gˆg(∇trˆgg− ∇K)·∇(|s|¯ ^2α_h ⁰)

≤CA|s|^α_h^0−βS

1 2

T ,ε+CA|s|^α_h^0−3β, 2 A|s|^2α_h ⁰

K−trgˆg|∇tr_ˆgg|²_g ≤ CA

K |s|^2(α_h ^0−β)S_T,ε ≤C₂A|s|^2α_h ⁰S_T,ε,

2 A|s|^2α_h ⁰ K−tr_ˆgg

Re

∇trˆgg·∇K¯

≤ 4A|s|^2α_h ⁰

K |∇trˆgg|g· |∇K|¯ g ≤ CA|s|^{2α0+2β−β−3β}S

1 2

T,ε

= CA|s|^4β_h S

1 2

T ,ε, and finally,

ρ²|s|^2α_h ¹ S_T,ε (K−tr_ˆgg)²

∆K

≤ρ²|s|^2α_h ¹S_{T ,ε} K²

C

|s|^4β_h ≤Cρ²|s|^2α_h ¹S_{T ,ε}

for some constants C, C₂ >0,C’s are different from each other in these estimates.

Since we may assume that |s|²_h <1, by choosing α₀ = 3β < α₁, we obtain at x₀, 0 ≥ −CA+ 4C1ρ²|s|^2(α_h ^1−β)

K² S_{T ,ε}

S_{T ,ε}− C 4C₁(S

1 2

T ,ε+ 1)− CK 4C₁S

1 2

T ,ε

+AC₁|s|^4β_h S_{T ,ε}−C₂A|s|^2α0S_T,ε−C₃AS

1 2

T ,ε−C₄S_{T ,ε} for some constants C, C₃, C₄ >0.

We may assume that S_T,ε > 1 at x₀ and then we may say that there exists a small constant κ >0 such that

S_{T ,ε}− C 4C1

(S

1 2

T ,ε+ 1)−CK 4C1

S

1 2

T,ε

> κ > 0.

And also we can say that, for sufficiently large A, β >0, we have at x₀,

|s|^4β_h

AC₁S_T,ε−C₂A|s|^2β_h S_T,ε− C₃A

|s|^4β_h S

1 2

T ,ε− C₄

|s|^4β_h S_{T ,ε}

> Aκ⁰²

(κ⁰⁰− C₄

Aκ⁰²)ST ,ε− C₃ κ⁰²S

1 2

T,ε

> Aκ⁰²

κ⁰⁰⁰S_{T ,ε}− C₃ κ⁰²S

1 2

T ,ε

since 0 < κ⁰ < |s|^2β_h (x₀) << 1 sufficiently small so that C₁ −C₂|s|^2β_h (x₀) > κ⁰⁰ > 0 by choosing a sufficiently largeβ for some small constant κ⁰⁰, κ⁰⁰⁰ >0 withκ⁰⁰−_Aκ^C402 > κ⁰⁰⁰ for sufficiently large A.. If κ⁰⁰⁰ST,ε−_κ^C02³S

1 2

T,ε≤0 at x0, we obtain S

1 2

T ,ε ≤ C₃ κ⁰⁰⁰κ⁰²,

hence we obtain the upper bound for S_{T ,ε} atx₀. On the other hand, if κ⁰⁰⁰S_{T ,ε}− _κ^C02³S

1 2

T ,ε >0 at x₀, we then have

|s|^4β_h (x₀)

AC₁S_{T ,ε}−C₂A|s|^2β_h S_{T ,ε}− C3A

|s|^4β_h S

1 2

T ,ε− C4

|s|^4β_h S_T,ε

(x₀)>0.

Therefore, at x₀, in this case we have ρ²|s|^2α1

K ST,ε≤ C₅A κ for some constant C₅ >0. Putting these together, we have f(x)≤f(x₀)≤

2ρ²(x₀) maxnC₅A κ ,|s|^2α_h ¹

K

C₃ κ⁰⁰⁰κ⁰²

2o

+C₆A|s|^2(α_h ^0−β)(x₀)

≤A(C₇+C₆) for some constant C₆, C₇ >0.

Hence, on B^r

2, we obtain

S_{T ,ε}≤ AC₈

|s|^2α_h ²

for some uniform constants C8, α2 :=α1 +β > 0. With using this computation for ST ,ε, we can obtain the upper bound for the curvature of ω_{T ,ε} and then we also have bounds on its all covariant derivatives by an analogue of [59, Proposition 4.2]. Additionally, with using the Sobolev inequality and a bootstrap argument, we obtain the higher order estimates.

We firstly show an estimate for its volume form and after that, with using the estimate, we can show estimates for ω_ε as in Lemma 4.2.2 by applying [59, Lemma 5.4] and [70, Lemma 3.5] respectively.

Lemma 4.2.3. ([59, Lemma 5.4])There exist positive constantsα andC, independent of ε, such that

ω_ε²

Ω_N ≤ C

|s|^2α_h on [T, T⁰]×(N \ {y₀}).

Lemma 4.2.4. For all sufficiently smallε >0, there exist positive uniform constants C, α, independent of ε, such that on [T, T⁰]×(N\ {y0}),

|s|^2α_h

C ωˆ_N ≤ω_ε≤ C

|s|^2α_h ωˆ_N.

Fix a large positive integer L. Then for each integer 0 ≤ k ≤ L there exist C_k, α_k > 0 such that

|(∇_R)^kω_ε|²_ω

T ,ε ≤ C_k

|s|^2α_h ^k

for t ∈ (T, T⁰], where ∇_R is the Levi-Civita covariant derivative associated to the metric ω_{T ,ε} =g.

Proof. We write ω_ε = √

−1(g_ε)_i¯jdzⁱ ∧ d¯z^j, ˆω_N = √

−1ˆg_i¯jdzⁱ ∧ d¯z^j and ˆω_t,N =

√−1(ˆg_t,N)_i¯jdzⁱ∧d¯z^j with local coordinates.

We define the quantity

Q⁰_ε := log(|s|^2α_h tr_gˆg_ε)−Aϕ_ε+ 1

˜ ϕε+C0

,

for sufficiently large α > 0, where ˜ϕ_ε := ϕ_ε− _A¹ log|s|^2α_h and choose a constant C₀ such that ˜ϕε+C0 ≥ 1, for A a large constant to be determined. Observe that Q⁰_ε tends to negative infinity as x∈ N tends to y₀, for any t ∈ [T, T⁰]. From Lemma 4.2.2, Q⁰_ε|_t=T is uniformly bounded from above by choosing α sufficiently large.

We apply [72, Proposition 3.1] to log trgˆgε, then we have ∂

∂t−∆_ε

log tr_gˆg_ε≤ 2

(tr_ˆgg_ε)²Re

g_ε^¯lkTˆ_kp^p ∇ˆ¯ltr_gˆg_ε

+Ctr_gεg,ˆ

where ∆_εis the Laplacian with respect tog_ε, ˆ∇is the covariant derivative with respect to ˆ

g, ˆT is the torsion tensor of ˆg, and assuming that we compute at a point where we have tr_ˆgg_ε ≥1. Suppose that Q⁰_ε achieves its maximum at x₀ ∈N\ {y₀}. Then we have atx₀,

∇ˆ¯ltr_ˆgg_ε

tr_ˆgg_ε =A∂¯lϕ˜_ε+ ∂¯lϕ˜_ε ( ˜ϕ_ε+C₀)² and with using this equality, we compute as in the estimate (f):

2 trgˆgε

Re

g_ε^¯lkTˆ_kp^p ∇ˆ¯ltr_ˆgg_ε trgˆgε

≤ |∂ϕ˜ε|²_gε

( ˜ϕε+C0)³ +CA²( ˜ϕ_ε+C₀)³ tr_gεgˆ (trˆggε)². We compute

∂

∂t −∆_ε

˜

ϕ_ε = log ω²_ε

Ω_N −tr_gε(g_ε−gˆ_t,N + α AR_h), where we used that √

−1∂∂¯log|s|²_h =√

−1∂∂¯logh away from y₀.

Since we may assume that at the maximum of Q⁰_ε we have (trgˆgε)² ≥ A²( ˜ϕε+C0)³, we have at x₀,

0 ≤ ∂

∂t −∆_ε Q⁰_ε

≤ |∂ϕ˜_ε|²_gε

( ˜ϕ_ε+C₀)³ +C⁰trgεˆg+

A+ 1

( ˜ϕ_ε+C₀)²

log Ω_N ω²_ε + 2

−

A+ 1

( ˜ϕε+C0)²

tr_gε(ˆg_t,N − α

AR_h)− 2|∂ϕ˜_ε|²_g

ε

( ˜ϕε+C0)³

≤ C⁰tr_gεgˆ+

A+ 1

( ˜ϕε+C0)²

logωˆ_N² ω²_ε −

A+ 1

( ˜ϕε+C0)²

tr_gε(ˆg_t,N − α AR_h) +

A+ 1

( ˜ϕ_ε+C₀)²

logΩ_N ˆ ω²_N

+ 2(A+ 1) for some uniform constantC⁰, C >0.

For an arbitrary fixed constant 1> c₀ >0, we have ˆ

g_t,N − α

AR_h ≥c₀gˆ_t,N

for any t ∈[T, T⁰] and for sufficiently large A > 0 and for all t∈ [T, T⁰]. If necessary, we again choose a much larger constant A, and then we have

Ac₀tr_gεˆg_t,N ≥(C⁰+ 1)tr_gεg.ˆ With using these estimates, we obtain atx₀,

0 ≤ C⁰tr_gεgˆ+

A+ 1

( ˜ϕ_ε+C₀)²

logωˆ_N²

ω_ε² −Ac₀tr_gεgˆ_t,N +

A+ 1

( ˜ϕ_ε+C₀)²

log Ω_N ˆ

ω²_N + 2(A+ 1)

≤ C⁰tr_gεgˆ+

A+ 1

( ˜ϕε+C0)²

logωˆ_N²

ω_ε² −(C⁰+ 1)tr_gεˆg+C Then at x0, we have

tr_gεˆg+

A+ 1

( ˜ϕ_ε+C₀)²

log ω²_ε ˆ

ω²_N ≤C

for some uniform constant C >0. Now we choose local coordinates around the point x₀ such that ˆg_i¯j(x₀) =δ_ij and (g_ε)_i¯j(x₀) = λ_iδ_ij with λ₁, λ₂ >0. Then we have

2

X

i=1

1 λ_i +

A+ 1

( ˜ϕ_ε+C₀)²

logλi

≤C.

Note that we have

A ≤

A+ 1

( ˜ϕ_ε+C₀)²

≤A+ 1.

For any λ >0, since the function λ7→ _λ¹ + (A+_{( ˜ϕ} ¹

ε+C0)²) logλ is uniformly bounded from below for sufficiently largeA, for each iwe have

1 λ_i +

A+ 1

( ˜ϕ_ε+C₀)²

logλ_i ≤C

for some uniform constantC >0. And then for eachi, we obtain (A+_{( ˜ϕ} ¹

ε+C0)²) logλ_i ≤C, which gives a uniform upper boundλ_i ≤C for some uniform constant C >0. Therefore, we have

tr_ˆgg_ε(x₀)≤C and

Q⁰_ε ≤Q⁰_ε(x₀)≤C

since ϕ_ε is uniformly bounded for all sufficiently small ε >0 as we see in Proposition 2.3.

Then we obtain, on [T, T⁰]×(N \ {y0}), ω_ε≤ C

|s|^2α_h ωˆ_N.

We can also obtain the bound of the Chern curvature tensor with respect to ω_ε, the bound of its covariant derivatives and the higher order estimates with the application of [45, Theorem 8.11.1&Theorem 8.12.1] by the same way as in [54].

Since ϕ_t=ϕ(t) for t∈[T, T⁰] is the limit ofϕ_ε as ε→0, the metric ω(t) = ˆω_t,N +√

−1∂∂ϕ¯ _t for t∈[T, T⁰] is a solution of the Chern-Ricci flow onN:

∂

∂tω(t) =−Ric(ω(t)) for t∈(T, T⁰], ω(T) = ω⁰.

Lemma 4.2.4 gives estimates on ω(t) for t ∈ [T, T⁰] on N \ {y₀} and Proposition 4.2.3 gives us estimates on ω(t) for t ∈ (0, T) on M \E. We can show that the Chern-Ricci flow can be smoothly connected at time T between [0, T)×M and (T, T⁰]×N, outside T × {y₀} ∼=T ×E via the map π.

Theorem 4.2.1. The solution ω(t) is a smooth solution of the Chern-Ricci flow in the space-time region R.

Proof. From Lemma 4.2.4 and Proposition 4.2.3, ω(t) satisfies the Chern-Ricci flow and is smooth at time T in the sense of Remark 4.1.1.

This completes the proof of (4) in Definition 1.2.5.

It remains to show that (N, ω(t)) converges in the Gromov-Hausdorff sense to (N, d_T) as t→T⁺. We obtain the following estimate by the same proof as in [59].

Proposition 4.2.4. ([59, Proposition 6.1]) There exist δ > 0 and a uniform constant C >0 such that for t∈[T, T⁰],

(1) ω(t)≤ _|s|^C2 h

ˆ ω_N, (2) ω(t)≤ ^C

|s|^2(1−δ)_h (π|⁻¹_M\E)^∗ω₀,

where ω₀ is the initial metric of the Chern-Ricci flow on M.

Proof. We identify a coordinate chart U aty₀ ∈N via coordinates (z₁, z₂) with the unit ball D in C²

D={(z₁, z₂)∈C²;|z₁|²+|z₂|² <1}.

Put r² :=|z₁|² +|z₂|². Letf_ε be a family of positive smooth functions onN of the form fε(z) = ε+r² on D, which converges to a function f which is of the form f(z) = r² on D and is positive on M \D. By the definition of the blow-down map, there is a smooth volume form Ω_M on M such that π^∗Ω_N = (π^∗f)Ω_M. Note that ˆω_t,N − _T^εωˆ_N is positive definite for sufficiently small ε >0 on N for t∈[T, T⁰]. For such sufficiently small ε >0, we consider the following family of Monge-Amp`ere flows on M:

∂

∂tρ_ε= log (π^∗(ˆω_t,N − _T^εωˆ_N) + _T^εω₀+√

−1∂∂ρ¯ _ε)²

(π^∗f_ε)Ω_M , ρ_ε|_t=T =ϕ_T−ε.

Observe that at t=T,

π^∗(ˆω_t,N − ε

Tωˆ_N) + ε Tω₀

t=T

= 1− ε

T

π^∗ωˆ_N + ε Tω₀

is equal to ˆω_T−ε, where ˆω_t is a family of reference metrics for t∈[0, T), of form ˆ

ω_t= 1

T((T −t)ω₀+tπ^∗ωˆ_N)∈[ω₀] +tc^BC₁ (K_M) since we haveπ^∗ωˆ_N =ω₀−TRic(ω₀) +√

−1∂∂u¯ ⁰₀ for a smooth function u⁰₀ onM. We can obtain a uniform bound for |ρ_ε|independent ofεby consideringρ_ε±A±(T−t) for sufficiently large uniform constantsA± >0 as in [61, Lemma 3.2]. For the upper bound of ρ_ε, we apply the maximum principle to

θ_ε,+:=ρ_ε+A₊(T −t)

for A₊ >0 a uniform constant to be determined later. Then we have

∂

∂tθ_ε,+ = log(π^∗(ˆω_t,N −_T^εωˆ_N) + _T^εω₀+√

−1∂∂θ¯ _ε,+)²

(π^∗f_ε)Ω_M −A₊.

SinceM×[T, T⁰] is compact,θ_ε,+attains a maximum at some point (x₀, t₀)∈M×[T, T⁰].

We claim that if A₊ is sufficiently large we have t₀ = T. Otherwise we have t₀ > T and then by applying Proposition 1.6 in [61], at (x0, t0),

0≤ ∂

∂tθε,+ ≤log(π^∗(ˆω_t0,N − _T^εωˆ_N) + _T^εω₀)²

(π^∗f_ε)Ω_M −A+≤ −1,

which is a contradiction, where we have chosen the uniform constantA so that A₊ ≥1 + sup

M×[T ,T⁰]

log(π^∗(ˆω_t,N −_T^εωˆ_N) + _T^εω₀)² (π^∗f_ε)Ω_M . Hence we have proved the claim that t₀ =T, which gives that

sup

M×[T ,T⁰]

θ_ε,+≤sup

M

θ_ε,+|_t=T = sup

M

ϕ_T−ε ≤C₊ for some uniform constantC₊ >0 and therefore

ρ_ε(x, t)≤A₊(t−T) +C₊ ≤A₊T⁰+C₊ for any (x, t)∈M ×[T, T⁰]. We apply a similar argument to

θε,− :=ρ_ε−A−(T −t) for A₋ >0 a uniform constant with

A− ≥1− inf

M×[T ,T⁰]log(π^∗(ˆω_t,N − _T^εωˆ_N) + _T^εω₀)² (π^∗f_ε)Ω_M .

Assume thatθε,− attains a minimum at some point (x₀, t₀)∈M×[T, T⁰] witht₀ > T and then we have at (x₀, t₀),

0≥ ∂

∂tθε,− ≥log(π^∗(ˆωt0,N − _T^εωˆN) + _T^εω0)²

(π^∗f_ε)Ω_M +A− ≥1, which is a contradiction. Hence we obtain t₀ =T and

M×[T ,Tinf ⁰]θε,− ≥inf

M θε,−|_t=T = inf

M ϕ_T−ε≥ −C−

for some uniform constantC− >0,

ρ_ε(x, t)≥ −A−T⁰ −C−

for any (x, t)∈M ×[T, T⁰], which gives the lower bound of ρ_ε.

And also, by modifying the argument in [59, Lemma 2.5] to deal with the extra terms coming from f_ε, we obtain, for

ω_ε :=π^∗ ˆ

ω_t,N − ε Tωˆ_N

+ ε

Tω₀+√

−1∂∂ρ¯ _ε, ω_ε ≤ C

|s|²_hπ^∗ω_N, ω_ε ≤ C

|s|^2(1−δ)_h ω₀

onM\E×[T, T⁰] andC^∞-estimates forωε on compact subsets away from E. By letting ε→0, and pushing forward toN, we obtain a smooth solution ˜ρof the following parabolic complex Monge-Amp`ere equation

∂

∂tρ˜= log(ˆω_t,N +√

−1∂∂¯ρ)˜ ²

Ω_N , ρ|˜t=T =ψT

onN \ {y₀} ×[T, T⁰] with ˆω_t,N +√

−1∂∂¯ρ˜satisfying the estimates (1), (2). On the other hand, ˜ρ is equal to the solution ϕ on N in Proposition 4.2.2. Hence, the estimates (1), (2) holds for ω(t).

Then we can obtain an analogue of [59, Lemma 2.6, Lemma 2.7] and then the conver-gence in the Gromov-Hausdorff sense follows by the argument in [59, Section 3].

Theorem 4.2.2. (N, ω(t)) converges in the Gromov-Hausdorff sense to (N, d_T) as t→T⁺.

Chapter 5

C ^α -convergence of the solution of the Chern-Ricci flow

on elliptic surfaces

5.1 A non-K¨ ahler properly elliptic surface

The normalized Chern-Ricci flow is given by







∂

∂tω(t) =−Ric(ω(t))−ω(t), ω(t)|_t=0 =ω₀,

where ω₀ =√

−1(g₀)_i¯jdzⁱ∧d¯z^j is a starting Gauduchon metric and the globally defined smooth real (1,1)-form locally given by

Ric(ω) =−√

−1∂∂¯log det(g) is the Chern-Ricci form of ω.

A non-K¨ahler properly elliptic surface M is a compact complex surface with its first Betti number b₁(M) = odd and the Kodaira dimension Kod(M) = 1 which admits an elliptic fibration π : M → S to a smooth compact curve S. The Kodaira-Enriques classification tells us that properly elliptic surfaces are the only one case for minimal non-K¨ahler complex surfaces with Kod = 1 (cf. [3, p.244]).

We assume that M is minimal, that is, there is no (−1)-curve on M. It has been shown that the universal cover of M is C×H [38, Theorem 28], where H is the upper half plane in C. Also, it is known that there is a finite unramified covering p:M⁰ →M which is a minimal properly elliptic surfaceπ⁰ :M⁰ →S⁰ andπ⁰ is an elliptic fiber bundle over a compact Riemann surfaceS⁰ of genus at least 2, with fiber an elliptic curve E (cf.

[12, Lemmas 1, 2]). So we firstly assume that π: M →S is an elliptic bundle with fiber E with genus g(S) ≥2, with M minimal, non-K¨ahler and Kod(M) = 1. That g(S)≥ 2 implies that the universal cover of S is the upper half plane H in C and there exists a metric onS with negative constant curvature induced by the Poincar´e metric onH, then we have c₁(S)<0. And also we have Kod(S) = 1.

It will be more convenient for us to work withC^∗×H, whereC^∗ :=C\ {0}. We define h:C×H →C^∗×H, h(z, z⁰) = (e^−z2, z⁰),

which is a holomorphic covering map. We will write (z₁, z₂) for the coordinates onC^∗×H and zi =xi+√

−1yi, xi, yi ∈R for i= 1,2, which means that we have y2 >0.

It has been shown by Maehara (cf. [46]) that there exists a discrete subgroup Γ ⊂ SL(2,R) with H/Γ = S, together with λ ∈ C^∗ with |λ| 6= 1 and C^∗/hλi = E and with a character χ : Γ→ C^∗ such that M is biholomorphic to the quotient of C^∗ ×H by the Γ×Z-action defined by

a b c d

, n

·(z₁, z₂) =

(cz₂+d)·z₁·λⁿ·χ a b c d

,az₂+b cz₂+d

for

a b c d

∈ Γ, n ∈ Z, and then the map π : M → S is induced by the projection C^∗ × H → H (cf. [6, Proposition 2], [75, Theorem 7.4]). Note that all orientation preserving isometries of the complex upper half planeH coincide with all linear fractional transformations of the form

z7→ az+b

cz+d with ad−bc= 1 for z∈H, a, b, c, d∈R. We define two forms on C^∗×H below:

α:=

√−1

2y₂² dz₂ ∧dz¯₂, γ :=√

−1(−2 z₁dz₁+

√−1

y₂ dz₂)∧(−2

¯ z₁d¯z₁−

√−1 y₂ d¯z₂).

The unique K¨ahler-Einstein metric ω_S onS with Ric(ω_S) =−ω_S is induced by the form α. Since we can check that the forms on C^∗×H;

√−1

y²₂ dz₂∧d¯z₂ and −_z²

1dz₁+

√−1

y2 dz₂ are Γ×Z-invariant, these forms α and γ are invariant under the Γ×Z-action. Hence they descend to M and we define a Hermitian metric discovered by Vaisman in [74].:

ω_V = 2α+γ,

which is a Gauduchon metric, i.e.,ω_V is a∂∂¯-closed Hermitian metric. Indeed, it satisfies that

∂ω¯ _V =−

√−1 y₂²z¯1

d¯z₂∧dz₂ ∧d¯z₁, ∂∂ω¯ _V = 0.

In [74, (2.9)], Vaisman introduced its pullback h^∗ωV by the holomorphic covering maph onC×H.

Note that we may work in a single compact fundamental domain for M in C^∗ ×H using z₁, z₂ as local coordinates and we may assume that z₁, z₂ are uniformly bounded and that y₂ is uniformly bounded from below away from zero.

Our main result is as follows:

Theorem 5.1.1. Let M be a minimal non-K¨ahler properly elliptic surface and let ω(t) be the solution of the normalized Chern-Ricci flow starting at a Gauduchon metric of the form

ω₀ =ω_V +√

−1∂∂ψ >¯ 0.

Then the metrics ω(t) are uniformly bounded in the C¹-topology, and as t→ ∞, ω(t)→π^∗ω_S,

in the C^α-topology, for every 0< α <1, where ωS is the orbifold K¨ahler-Einstein metric on_√ S with Ric(ω_S) = −ω_S away from finitely many orbifold points induced by the form

−1

2y² dz∧dz¯on C^∗×H,H is the upper half palne in C, z ∈H is the variable,y= Imz.

5.2 Proof of Theorem 5.1.1

We define reference metrics

˜

ω :=e^−tω_V + (1−e^−t)α=e^−tγ+ (1 +e^−t)α,

which are Hermitian metrics for any t ≥ 0. We denote these metrics ˜g and also denote quantities with respect to ˜g with using a tilde such as the torsion tensor, the Chern connection and the Chern curvature tensor.

We define a volume form Ω by

Ω = 2α∧γ

and we write Ric(Ω) for the globally defined real (1,1)-form given locally by−√

−1∂∂¯log Ω.

Then we have

Ric(Ω) =−α∈c^BC₁ (M) = −c^BC₁ (KM), .

which implies that c^BC₁ (M) = π^∗c₁(S). Since we have assumed that g(S) ≥ 2, we have c₁(S)<0. So we have c^BC₁ (K_M)≥0, which means that the first Bott-Chern class of the canonical bundle c^BC₁ (K_M) is nef. Here, we say that c^BC₁ (K_M) is nef if for any ε > 0, there exists a real smooth function f_ε on M such that −Ric(ω₀) + √

−1∂∂f¯ _ε > −εω₀, or equivalently for any ε > 0, there exists a smooth Hermitian metric h_ε on the fibers of the canonical bundle K_M with its curvature form bigger than −εω₀. Hence from [71, Theorem 2.1], the normalized Chern-Ricci flow (equivalently the Chern-Ricci flow) has a smooth solution defined for all t ≥0. For instance, the following time-metric scaling for the solution of the Chern-Ricci flow







∂

∂tω(t) = −Ric(ω(t)), ω(t)|_t=0 =ω₀,

allows us to transform a solution of the normalized Chern-Ricci flow:

ω(t) =e^sω(s),˜ s(t) = log(t+ 1),







∂

∂sω(s) =˜ −Ric(˜ω(s))−ω(s),˜ ω(s)|˜ _s=0 =ω₀,

ドキュメント内 ON THE CONVERGENCE OF THE CHERN-RICCI FLOW ON COMPLEX SURFACES (ページ 63-110)